Question

Find the HCF of \[455\] and \[84\] by Euclid’s division lemma.

Answer 1

Hint: Here we will be using the formula of Euclid’s division which states that if we have any two positive integers \[x\] and \[y\], then there would be whole numbers \[q\] and \[r\] that will satisfy the equation
\[x = yq + r\], where \[0 \leqslant r \leqslant y\]. Here, \[x\] represents the dividend and \[y\] represents the divisor.

Complete step-by-step solution:
Step 1: We will do the factorization \[455\] by using Euclid’s division lemma formula as shown below:
\[455 = 84 \times 5 + 35\]
Where,
\[x = 455\],
\[y = 84\],
\[q = 5\] and
\[r = 35\].
Step 2: Now by considering the divisor as \[84\] and the remainder as \[35\]we will again apply Euclid’s division lemma rule as shown below:
\[ \Rightarrow 84 = 35 \times 2 + 14\]
Similarly, again repeat the same step by considering \[35\] as a divisor and \[14\] as remainder:
\[ \Rightarrow 35 = 14 \times 2 + 7\]
We will again repeat the same step till the remainder will be equals to zero. Now, consider \[14\] as the divisor and \[7\] as remainder:
\[ \Rightarrow 14 = 7 \times 2 + 0\]
Step 3: Thus, the last divisor we get after applying Euclid’s rule is \[7\]. So, the HCF will be equal to \[7\].

The HCF of the number \[455\] and \[84\] is \[7\].

Note: Students should remember that while calculating the HCF (Highest common factor) between two positive integers by using Euclid’s division lemma rule, we will repeat the formula until the remainder is zero.
There are different methods for calculating the HCF as mentioned below:
Factorization method
Prime factorization method
Division method